Post on 04-Jun-2018
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
1/139
ABSTRACT
Title of Document: BAYESIAN BELIEF NETWORK AND FUZZY
LOGIC ADAPTIVE MODELING OF DYNAMIC
SYSTEM: EXTENSION AND COMPARISON
Ping Danny Cheng, M.S., 2010
Directed by: Professor Mohammad Modarres, Mechanical
Engineering Department
The purpose of this thesis is to develop, expand, compare and contrast two
methodologies, namely BBN and FLM, which are used in the modeling of the
dynamics of physical system behavior and are instrumental in a better understanding
on the POF. The paper begins with an introduction of the proposed approaches in the
modeling of complex physical systems, followed by a quick literature review of FLM
and BBN. This thesis uses an existing pump system [3] as a case study, where the
resulting NPSHA data obtained from the applications of BBN and FLM are compared
with the outputs derived from the implementation of a Mathematical Model. Based on
these findings, discussions and analyses are made, including the identification of the
respective strengths and weaknesses posed by the two methodologies. Last but not
least, further extensions and improvements towards this research are discussed at the
end of this paper.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
2/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
3/139
ii
Acknowledgements
I would first like to thank Professor Mohammad Modarres for his guidance and
advice towards the development of my thesis.
I would also like to thank my wife, Pristine, for her support, love and company during
the difficult times.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
4/139
iii
Table of Contents
Acknowledgements ........................................................................................................ ii
Table of Contents ......................................................................................................... iii
List of Tables ................................................................................................................. v
List of Figures ............................................................................................................... vi
Chapter 1: Introduction .................................................................................................. 1
Chapter 2: Proposed Approaches in Modeling Complex Physical System ................... 4
2.1 Logic Based Illustration of Physical Systems ...................................................... 6
2.2 Illustration of Fuzzy Logic Model ....................................................................... 8
2.3 Illustration of Proposed Bayesian Belief Network Model ................................. 10
Chapter 3: Fuzzy Logic Modeling ............................................................................... 14
3.1 Fuzzification ....................................................................................................... 16
3.2 Defuzzification Methods .................................................................................... 17
3.2.1 Center of Area Defuzzification.................................................................... 17
3.2.2 Center of Sums Defuzzification .................................................................. 18
3.2.3 Mean of Maxima Defuzzification ................................................................ 19
Chapter 4: Bayesian Belief Network ........................................................................... 20
Chapter 5: Applications of Proposed Bayesian Belief Network and Fuzzy Logic
Models.......................................................................................................................... 23
5.1 Mathematical Model .......................................................................................... 25
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
5/139
iv
5.2 Fuzzy Logic Model ............................................................................................ 26
5.3 Proposed Bayesian Belief Network Model ........................................................ 35
Chapter 6: Findings and Discussion ............................................................................ 52
6.1 Discussion on Bayesian Belief Network Probability Data ................................. 53
6.2 Comparison of NPSHA1 Data with Reference Data ......................................... 56
6.2.1 Bayesian Belief Network Model Comparison and Discussion ................... 56
6.2.2 Fuzzy Logic Comparison and Discussion ................................................... 66
6.2.3 Comparison between Bayesian Belief Network and Fuzzy Logic Model ... 73
Chapter 7: Conclusion and Recommendation.............................................................. 80
7.1 Conclusion .......................................................................................................... 80
7.2 Recommendations .............................................................................................. 81
Appendix A .................................................................................................................. 84
Appendix B .................................................................................................................. 88
Appendix C .................................................................................................................. 92
Bibliography .............................................................................................................. 129
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
6/139
v
List of Tables
Table 1: Generalized representation of CPT of output Y ............................................ 13
Table 2: Some Fuzzy Implication Operators [9].......................................................... 16
Table 3: Interpretation of ELSE under some Implication [9] ...................................... 17
Table 4: Temperature and GPM Category breakdown ................................................ 27
Table 5: Probability data and value of A to L based on the CDF of NPSHA1 ........... 38
Table 6: Probability data of GPM and Temperature inputs ......................................... 38
Table 7: Representation of GPM and Temperature variables ...................................... 39
Table 8: Numerical values of A to L ........................................................................... 40
Table 9: Mean and Standard Deviation of 12 sets ....................................................... 43
Table 10: Probability of A to L .................................................................................... 51
Table 11: Comparison of probability data of NPSHA1 based on 2 scenarios ............. 55
Table 12: Comparison of NPSHA1data between BBN model and Reference model at
T = 40, 80, 120 and 160 ͦ F ........................................................................................... 59
Table 13: Comparison of NPSHA1data between Fuzzy Logic model and Reference
Model at T = 40, 80, 120 and 160 ͦ F ........................................................................... 69
Table 14: Table of Comparison between BBN and Fuzzy Logic Models ................... 75
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
7/139
vi
List of Figures
Figure 1: Logic Based Illustration of Physical Systems ................................................ 6
Figure 2: Fuzzy Logic Control Analysis Method [8] ..................................................... 8
Figure 3: Membership function of Input parameter θ with overlaps ............................. 9
Figure 4: Illustration of Proposed BBN Method.......................................................... 11
Figure 5: Bayesian Network over five propositional variables [7] .............................. 21
Figure 6: A Pumping System [3] ................................................................................. 24
Figure 7: DMLD for simulating NPSHA [3] ............................................................... 24
Figure 8: Numerical representation of NPSHA vs. GPM ............................................ 25
Figure 9: Rules based between Temperature and GPM ............................................... 26
Figure 10: Fuzzy logic illustration of Pump System at Z0 = 0 ..................................... 26
Figure 11: FIS interface with 2 input and 1 output parameter ..................................... 28
Figure 12: Membership function of Temperature ........................................................ 28
Figure 13: Membership function of GPM ................................................................... 29
Figure 14: Membership function of output NPSHA1 .................................................. 29
Figure 15: Rules conditions between input and output parameters ............................. 30
Figure 16: 3D Surface view of NPSHA1 ..................................................................... 31
Figure 17: 2D Surface view of NPSHA1 with respect to GPM .................................. 31
Figure 18: Pump GPM vs. NPSHA1 ........................................................................... 32
Figure 19: FIS of NPSHA with Z0 parameter .............................................................. 33
Figure 20: Membership function of Z0 ........................................................................ 33
Figure 21: Membership function of NPSHA1 at Temp = 0 ........................................ 34
Figure 22: Membership function of NPSHA at Temp =0 ........................................... 34
Figure 23: Surface view of NPSHA at T=0 with respect to NPSHA1 and Z0 ............. 35
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
8/139
vii
Figure 24: Normal distribution for input Temperature and GPM ................................ 36
Figure 25: PDF and CDF of NPSHA1output .............................................................. 36
Figure 26: Histogram of NPSHA1 to estimate the probability of A to L .................... 37
Figure 27: BBN interpretation of the pump system ..................................................... 41
Figure 28: Conditional Probability table assuming no uncertainty between A to L .... 42
Figure 29: PDF of Set 1 which represents the GPM_HI and Temp_HI ...................... 45
Figure 30: CDF of Set 1 which represents the GPM_HI and Temp_HI ...................... 45
Figure 31: PDF of set 2 which represents the GPM_MH and Temp_HI..................... 46
Figure 32: CDF of set 2 which represents the GPM_MH and Temp_HI .................... 47
Figure 33: PDF of set 1 which represents the GPM_ZE and Temp_LW .................... 48
Figure 34: CDF of set 12 which represents the GPM_ZE and Temp_LW .................. 49
Figure 35: Conditional probabilities of A to L ............................................................ 49
Figure 36: Distribution of NPSHA given GPM and Temperature evidence ............... 56
Figure 37: BBN of pump system given evidence that NPSHA1 is D ......................... 61
Figure 38: Dynamic Bayesian Network with feedback loop [11] ............................... 62
Figure 39: Time expansion of the dynamic network in Figure 34 [11] ....................... 63
Figure 40: Training module within a SD environment [16] ........................................ 64
Figure 41: Comparison of NPSHA at T=40 between Reference model and FLM ...... 67
Figure 42: NPSHA1 comparison based on the reference model, FLM and BBN at
Temp = 40. ................................................................................................................... 73
Figure 43: NPSHA1 comparison based on the reference model, FLM and BBN at
Temp = 160 .................................................................................................................. 74
Figure 44: Example of Continuous BBN where each node is a continuous chance
node .............................................................................................................................. 82
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
9/139
1
Chapter 1: Introduction
Uncertainties within data or information are inherent in complex dynamic
systems, underscoring the challenges faced in the development of dynamic models.
Empirical information may be non-existent or are not easily available in physical
systems; therefore it is not uncommon to fall back on expert opinions as the main
source of information. With respect to these issues, there is a need to identify more
simplified methodologies to model complex physical system behaviors, in particular,
the dynamics of systems that support the POF. In this regard, two methodologies have
been identified: FLM and BBN.
Probability theory is synonymous with the modeling of stochastic uncertainty,
which deals with the uncertainty of the occurrence of a specific event. BBN which is
also known as causal belief network [1] prescribes to the probabilistic model. It is a
powerful tool that can be used to model a wide variety of domains, which includes
diagnosis of electronic/mechanical systems, ecosystem and organizational factors.
On the other hand, fuzzy logic involves a tradeoff between precision and
significance. It represents uncertainty via fuzzy sets and membership function [2].
Fuzzy logic and probabilistic logic are mathematically similar where both have truth
values ranging between 0 and 1. One significant difference is that fuzzy logic focuses
on the degrees of truth, while probabilistic logic revolves around probability and
likelihood.
The fuzzy set theory explains day-to-day realities better than the probability
theory because not all phenomena and observations assume only two definite states.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
10/139
2
However, the modern and methodical science of fuzzy logic is still in its budding
stage and warrants further research before more definitive conclusion can be provided
[2].
In light of this, this thesis attempts to adopt the FLM and the BBN model into
the analyses of complex physical systems. The objective of this thesis is to develop,
expand, compare and contrast these two methodologies. An application of the use of
these two methods is also developed to better understand their strengths and
weaknesses. The application that was used in this thesis was adapted from the pump
system example of an earlier PhD. research conducted by S.H. Hu at the University of
Maryland [3].
Proposed BBN and FLM approaches in modeling complex physical system are
discussed in Chapter 2. Theoretical background studies on FLM and BBN are
presented in Chapter 3 and 4. Chapter 5 describes the implementation of the BBN and
FLM models on the pump system, where the procedure to obtain the NPSHA output is
also clearly defined. For this comparison study, NPSHA is estimated based on the
assumption that Z0 is equal to zero. Chapter 6 reports on the results obtained from
both the BBN and FLM, followed by an in-depth discussion on the advantages and
disadvantages of the two models. Chapter 7 looks into possible extensions of the two
methodologies on more complex systems, suggests future research that could be
conducted, and rounds up the thesis in the conclusion.
The main contributions of this thesis are: 1) to propose a methodology
applicable to BBN in estimating behavior of complex physical systems; 2) to adopt
tools to compute complex BBN based on the proposed methodology; 3) to automate
the solution of S.H. Hu’s [3] FLM approach in modeling complex physical system
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
11/139
3
behavior; and 4) to compare, contrast and assess the accuracy and uncertainty of the
two methods.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
12/139
4
Chapter 2: Proposed Approaches in Modeling Complex Physical
System
There are several different methods to define complex physical systems.
According to Marashi and Davis, a complex physical system contains many
components and layers of subsystems with multiple non linear interconnections that
are difficult to recognize, manage and predict [4]. Solidova and Johnson also
highlighted that it is difficult to predict complex time dependent changes within
interactions of components or subsystems, in response to rapidly changing properties
of both systems and environment [5].
Due to the complex nature of dynamic systems, mathematical models are often
used to produce numerical results that represent some observable aspects of system
behaviour in the physical sciences or engineering disciplines [6]. It would be ideal if
such methodologies are readily available to model complex physical systems
behaviors, or are straightforward and easy to work on. However in reality, this is
rarely the case. Mathematical models are usually based on complicated concepts such
as higher order/partial differentiation which can be time consuming, and the intricate
computations required may pose great difficulties for novices in solving complex
system problems. Therefore, the use of mathematical models in the industrial context
may be constrained by limited resources available, as the complexities involved in
these models require hiring of mathematical experts or purchasing of relevant
software programs tailored to the specific needs of the mathematical model, which
could be non-economical for most practitioners.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
13/139
5
Another reason for advocating the use of simplified methodologies in place of
mathematical models was mentioned in Chapter 1, where empirical information may
be non- existent or not easily available in physical complex systems. In such cases,
expert opinions comprising of uncertainties, variations and subjective judgments will
form the best alternative source of information, reinforcing the strengths of using
BBN and FLM to solve complex system problems. . On the other hand, mathematical
models tend to be more rigid and inflexible, as they are unable to account for such
uncertainty and variability. Any variation or change to the data might result in
disproportionate changes in the computations of results by the mathematical models,
which may not reflect the actual impact of the changes.
Instead of turning to complex mathematical models, there is therefore a need
to search for more simplified methodologies that require lesser time and resources to
model complex physical system behavior. In the context of this thesis, two
methodologies, FLM and BBN have been identified as simpler alternatives to
mathematical models, where both can be represented graphically in providing more
direct platforms for analyses, as opposed to working with complicated mathematical
equations. FLM and BBN are reliable and yet more time-efficient methods, as they do
not require exact historical data or evidence to produce convincing results. Both
proposed models are also able to account for variability and uncertainty of input and
output data, where such flexibility is lacking in conventional mathematical models.
To recap on the concepts of the proposed methodologies, FLM represents uncertainty
via fuzzy sets and membership function [2], while BBN epitomizes probabilistic
dependency models that represent random stochastic uncertainty via its nodes [7].
This chapter first discusses the logic based illustration of generic physical
systems and how it can be represented in the form of matrices. This is followed by a
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
14/139
6
discussion on the frameworks and use of FLM and BBN in solving the dynamics of
these complex physical systems. In essence, this chapter explains the core
fundamentals for the implementation of both methodologies on generic physical
systems before the thesis focuses on a specific example of a complex system, the
pump system case study.
2.1 Logic Based Illustration of Physical Systems
The interaction between the inputs and outputs of physical systems can be
represented by matrices that are made up of dependent and independent
variables/parameters. These variables and parameters may be divided into distinct
ranges that have their own unique features and functions. These ranges are not
arbitrary and can be represented either quantitatively or qualitatively. The division of
each range has physical meaning and could result in phenomenal changes or a shift in
the rate of change of dependent variables i.e., A shift from a gradual slope to a steep
slope. For a more explicit illustration of a physical system, refer to Figure 1 below:
Figure 1: Logic Based Illustration of Physical Systems
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
15/139
7
Figure 1 shows a logic based illustration of physical systems that is made up
of multiple equations. The general mathematical model of the physical system can be
represented by Y = f (θ, X, Z), where Y is a matrix of discrete vector input parameters
θ and variables X and Z.
θ , X , Y and Z may be divided into distinct ranges and can be represented as follows:
θ = {θ , θ , θ , θ … , θ
X = {X1, X2 …, Xn}
Y = {Y1, Y2, Y3, Y4, Y5 …, Yn}
Z = {Z1, Z2 …, Zn}
The lattice is made up of a system of equations:
Y2 = f (θ2, X2), Y3 = f (θ3, X2, Z2), Y4 = f (θ4), Y5 = f (θ3, Z1), Yn = f (θn, Xn, Zn)
Note that Z1 is an input to X1, which is an input to Y2; and Z2 is an input to θ3 which
is an input to Y3 and Y5.
In the event that mathematical models are not available, or require too much
resource to solve the relevant system problems, the interaction between the lattices
can be developed by expert judgment either through FLM or BBN, as a more
effective alternative. The implementation of the two models on complex physical
systems will be discussed shortly in the following sections.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
16/139
8
2.2 Illustration of Fuzzy Logic Model
Figure 2: Fuzzy Logic Control Analysis Method [8]
Figure 2 shows an overview of how a complex physical system can be
represented as a FLM. The ranges of the input and output parameters/variables are
represented by membership functions and fuzzy sets. In addition, the interactions
between input and output variables/parameters are represented by fuzzy rules. In a
nutshell, system input parameters and variables are encoded into fuzzy representations
using well defined “If/Then” rules which are converted into their mathematical
equivalents. These rules would then determine actions to be taken based on
Implication Operators such as Zadeh Min/Max, or Mamdani Min. The fuzzified data
is then put through a defuzzification process via Center of Area, Center of Sum or
Mean of Maxima methods to obtain a crisp output value.
In order to better explain how FLM can be implemented into a complex
physical system; refer back to the illustration of physical systems as shown in Figure
1. The input and output parameters/ variables of the physical system, θ, X, Y, and Z
go through a fuzzification process ( Refer to Figure 2).
Input
Measurement
or assessment
of system
parameters and
variables
i.e., θ, X, Y, Z
Fuzzification
Using human determined
fuzzy “If/Then” rules to
determine actions to be
taken based on
Implication Operator
I.e. Mamdani min
Defuzzification
Methods: COA,
COS or MOM.
Goal is to determine
the centre of mass
for all system
conditions
(Averaging)
Output
Crisp Behavior
Data
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
17/139
9
Figure 3 shows a simple graphical illustration of the membership function of
the input parameter θ. The fuzzy sets determine the different grades of the
membership function that is made up of distinct ranges of θ1, θ2, θ3 … θn. Therefore,
the fuzzy set of θ1 can be set between the interval θ1min and θ1max; the fuzzy set for
θ2 can be set between the interval θ2min and θ2max; and the fuzzy sets for the
remaining membership functions can be assigned accordingly. The input parameters θ
take the form of triangular shaped membership functions. Note that the membership
functions allow overlaps between the members which accounts for the approximations
and uncertainties between the parameters/variables. The same steps can be taken to
fuzzify variables X, Z and output Y.
Figure 3: Membership function of Input parameter θ with overlaps
The fuzzy rules of the physical system as shown in Figure 1 can be
represented via the “If/Then” rules as follows:
if θ = θ2 and X = X1 then Y = Y2 ELSE
if θ = θ3 and X = X2 and Z = Z2 then Y= Y3 ELSE
if θ = θ4 then Y= Y4 ELSE
1
0 θ1min θ2min θ1max θ3min θ2max θ4min θ3max θ4max
θ1 θ2 θ3 θn
Input θ
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
18/139
10
if θ = θ2 and Z = Z1 then Y = Y2 ELSE
if θ = θn and X = Xn and Z = Zn then Y= Yn ELSE
if Z = Z1 then X= X1 ELSE
if Z = Z2 then θ = θ3 ELSE
For this example, the implication operator, ϕ is Mamdani min. Φµx, µBy µx µBy, where μA and μB are membership functions of Aand B, and its interpretation for ELSE is AND (
). Section 3.1 would further discuss
some of the other fuzzy implication operators that can be used.
The defuzzification method uses COA to determine the centre of mass for all
system conditions in order to obtain crisp output data. The COA methodology can be
found in section 3.2.1 and would be further explained. Section 3.2 would look into
some other defuzzification methods that can be used to obtain a crisp output data.
2.3 Illustration of Proposed Bayesian Belief Network Model
This section illustrates how complex physical system can be represented by
the proposed BBN model. Either discrete probability or continuous probability
method can be employed to estimate the output Y. However, the latter method is a
better option to solve the proposed BBN modelling as there is a need to account for
the uncertainties and overlaps between input/output intervals of a system.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
19/139
11
Figure 4: Illustration of Proposed BBN Method
Refer to Figure 4 where the focus of this illustration is on output Y. Assume
that parameter θ1 and variable X1 would lead to an output YA represented by a right
truncated normal distribution labelled “A” that ranges between the interval Y1min and
Y3max. EY1 is the probability that the output Y falls within the interval Y1, which is
between limits Y1min and Y1max. Similarly, assume that the input θ2 and X1 would lead
to an output YB represented by a normal distribution labelled “B” that ranges between
θ X
Y
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
20/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
21/139
13
EY Pr Y | θ θ θ X X X PrY | θ θ θ X X X PrY | θ θ θ X X X
Note that the summation of EY1, EY2 … EYo must add up to one. Table 1
shows a generalized representation of the CPT of output Y:
Table 1: Generalized representation of CPT of output Y
The methodologies of the proposed approaches in modeling complex physical
system that was described in this chapter would be further discussed in the next
chapter.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
22/139
14
Chapter 3: Fuzzy Logic Modeling
Fuzzy systems represent a unique approach to represent uncertainties that
usually arise from complex systems. As quoted by Lotfi A. Zadeh, “As the complexity
of a system increases, it becomes more difficult and eventually impossible to make a
precise statement about its behavior, eventually arriving at a point of complexity
where the fuzzy logic method born in humans is the only way to get at the
problem.” [8].
A fuzzy system is deterministic and time invariant where the input and output
parameters are encoded in fuzzy representations and the interrelationships between
the fuzziness take the form of well defined if/then rules. The fuzzy system then
converts these rules to their mathematical equivalents, which would simplify the
interaction between the human and computer. This in turn offers a more realistic and
accurate representation of system behavior in the real world.
Fuzzy logic deals with reasoning that hinges on approximation rather than
precision. This presents a stark contrast to crisp logic where binary sets have binary
logic and the logic variables have a membership value of either 0 or 1.
The Fuzzy Logic Toolbox [2] can be used to create a fuzzy logic system. This
toolbox is a collection of functions built on the MATLAB numeric computing
environment which enables one to create and edit fuzzy inference system within the
MATLAB interface. The implementation of this function with an existing application
will be discussed in Chapter 5 of this paper.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
23/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
24/139
16
3.1 Fuzzification
Fuzzy if/then rules are conditional statements that describe the dependence of
one or more linguistic variables on another. The underlying analytical form of the
fuzzy if/then rule is called the implication relation [9]. These relations are obtained
through different fuzzy implication operator Φ. Information from the LHS and RHS
of a rule is imputed to Φ, and it outputs an implication relation. Note that μ
represents the membership function of n [9].
For example [9], consider an if/then rule involving two linguistic variables:
if x is then y is ELSE
if x is then y is ELSE …
if x is then y is
where the linguistic variable x (LHS) and y (RHS) takes the value and respectively.
Table 2: Some Fuzzy Implication Operators [9]
Name Implication Operator
, , Zadeh Max-Min 1
, Mamdani min ,Larsen Product . , Arithmetic 1 1
, Boolean 1
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
25/139
17
, Bounded Product 0 1
Table 3: Interpretation of ELSE under some Implication [9]
Implication Interpretation of ELSE
, Zadeh Max-Min AND ()
, Mamdani min OR (),Larsen Product OR ()
, Arithmetic AND (), Boolean AND ()
, Bounded Product OR ()
3.2 Defuzzification Methods
Defuzzification [9] is a process of selecting a crisp number u* representation
from the membership function output µ . This step takes place after the inputs to thecontroller has been processed by the fuzzy algorithm. The most commonly used
defuzzification methods are COA, COS, and MOM.
3.2.1 Center of Area Defuzzification
The crisp value u* is taken to be the
geometrical center of the output fuzzy
value μ , where μ is the union of all
the contributions of rules whose DOF >
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
26/139
18
0.
u ∑ u μ u∑ μ u
Where is the universe of discourse and N is the number of samples. It is a
commonly used defuzzification method,
and is also known as Centroid.
3.2.2 Center of Sums Defuzzification
Easy to implement, and has fast
inference cycle. COS takes into
account the overlapped areas of
multiple rules more than once.
COS takes the sum of the outputs
from each contributing rule and not
just the union.
∑ ∑ ′ ∑ ∑ ′
Where
′
is the membership
function resulting from firing the kth
rule.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
27/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
28/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
29/139
21
• Able to display variables in a model as nodes in a network, and causes and
effects as links between the nodes.
• Able to diagnose current situation based on past data.
An example of a BBN extracted from Adnan Darwiche’s paper [7] is shown below:
Figure 5: Bayesian Network over five propositional variables [7]
Figure 5 shows a BN with five nodes, Z = {A, B, C, D, E}. The five tables are
known as CPT ΘB| where it denotes the CPT for variable B, and its parent A. θ| is
used to denote the value assigned by the CPT ΘB| to the conditional probability Pr
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
30/139
22
(b|a). Note that the sum of θ| must add up to one. In addition, conditional
probabilities represent the likelihoods based on prior or historical information.
Based on the above BN, the probability of winter being true given the
conditions that sprinkler is on; there is no rain; grass is wet; and road is not slippery is
as follows:
Pr , , , , ) = θ θ | θ | θ |, θ |
= (0.6) * (0.2) * (0.2) * (0.9) * (1) = 0.0216
Similarly, the probability that winter is false, given the conditions that
sprinkler is off, there is no rain, grass is not wet and road is not slippery is as follows:
Pr , , , , ) = θ θ | θ | θ |, θ |
= (0.4) * (0.25) * (0.9) * (1) * (1) = 0.09
Further explanations on the terminology of the BBN can be found in Appendix B.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
31/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
32/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
33/139
25
5.1 Mathematical Model
Consider the system where 100 and the pipe is 4 inches in diameter.
For a given relative installation location of a pump-sink set, the NPSHA can be
expressed as:
NPSHA 35.18 Z 6.4 10 GPM 0.085 Temperature ͦ F [3]
The pump GPM ranges between 0 and 480 and temperature falls in the range
of 0 to 200 ͦ F. Note that Z = 0 would give the worst case scenario for NPSHA at anygiven GPM and Temperature data. Thus to simplify this application, Z is assumed to
be zero.
Figure 8: Numerical representation of NPSHA vs. GPM
NPSHA’s output calculation of six different temperature ranges was based on
the physical model. These data is used as the reference data for comparison between
the FLM and the proposed BBN Model. Figure 8 captures the plot of GPM vs.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
34/139
26
NPSHA when = 0. As the distance between pump and water surface increases, NPSHA adjusts according to the increase in .
5.2 Fuzzy Logic Model
The first step of developing the FLM is to define the rules ( Figure 9) based on
the input conditions mapped by the DMLD as shown in Figure 7. The fuzzy logic
illustration of the Pump System when = 0 is shown in Figure 10.
GPM
HI MH MD ML LW ZE
TEMP
HI A B C D E F
LW G H I J K L
Figure 9: Rules based between Temperature and GPM
Figure 10: Fuzzy logic illustration of Pump System at = 0
G1 G2 G3 G4 G5 G6
T1 A B C D E F
T2 G H I J K L
NPSHA1
Temp GPM
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
35/139
27
Assumptions made in the application of this model are shown in Table 4.
Table 4: Temperature and GPM Category breakdown
Temperature Range Category Symbol
0 to 100 deg F Low TempLW
101 to 200 deg F High TempHI
GPM Range Category Symbol
-40 to 40 Zero ZE
0 to 120 Low LW
80 to 200 Mid Low ML
160 320 Mid MD
280 to 400 Mid High MH
360 to 480 High HI
MATLAB’s FLT function was used in this instance where the FIS structure is
a MATLAB object that contains all the fuzzy inference system information.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
36/139
28
Figure 11: FIS interface with 2 input and 1 output parameter
Figure 12: Membership function of Temperature
The input Temperature is made up of a Z-shaped and S-shaped membership
function.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
37/139
29
Figure 13: Membership function of GPM
The GPM inputs are made up of triangular-shaped membership functions, and
the Z/S shaped membership functions at the extreme ends.
Figure 14: Membership function of output NPSHA1
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
38/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
39/139
31
The surface view of the crisp NPSHA1 is shown in the figure below.
Figure 16: 3D Surface view of NPSHA1
Figure 17: 2D Surface view of NPSHA1 with respect to GPM
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
40/139
32
Figure 18: Pump GPM vs. NPSHA1
Figure 18 shows the visualization of the NPSHA1 output with respect to
Temperature and GPM. From the surface view of NPSHA1, GPM vs. NPSHA (=0)is plotted and compared with the reference plot obtained via the mathematical model.
The trends of the graphs at all 6 temperature points are consistent with the reference
plot as shown in Figure 8. It is not possible to obtain precise output solution as FLM
is based on approximation given limited input and output data. In order to obtain a
smoother curve with higher resolution, the membership functions of the input/output
parameters needs to be broken down into more defined categories, and fuzzy rules
need to be defined with greater accuracy.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
41/139
33
The output of NPSHA can be estimated by incorporating with NPSHA1using the FIS controller as shown in Figure 19. NPSHA output can be estimated by
using the same methodology to estimate the initial NPSHA1.
Figure 19: FIS of NPSHA with parameter
Figure 20: Membership function of
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
42/139
34
The membership function of is grouped into either positive or negativecategory. Similarly, the membership function of NPSHA1 at individual temperature is
grouped according to five categories: NE, SN, NT, SP and PT. For this application,
the area of focus would be on NPSHA1 when temperature is zero. The fuzzy
NPSHA1 output range is shown in Figure 21, when T=0 is between 21 and 36, and
the triangular membership function is distributed across the output range.
Figure 21: Membership function of NPSHA1 at Temp = 0
Figure 22: Membership function of NPSHA at Temp =0
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
43/139
35
The fuzzy output of NPSHA is then represented by the triangular membership
function labeled M to V as shown in Figure 22
Figure 23: Surface view of NPSHA at T=0 with respect to NPSHA1 and Z0
Referring to the surface view of output in Figure 23, the trend reveals that
NPSHA increases proportionally with Z0 increase. This estimated result is reasonable
given that the result obtained from the mathematical model in Section 5.1 is similar,
where NPSHA also exhibits a proportional increase when Z0 increases.
The same steps were repeated to obtain the corresponding fuzzy outputs of
NPSHA at temperatures of 40 ͦ F, 80 ͦ F, 120 ͦ F, 160 ͦ F and 200 ͦ F respectively.
5.3 Proposed Bayesian Belief Network Model
Similar to the fuzzy logic method, BBN methodology adopts a probabilistic
approach to estimate the output NPSHA. Consider the case where inputs of the system
follow a normal distribution. To solve NPSHA1, is assumed to be zero. UsingMonte Carlo simulation for a sample size of 5000, both the inputs Temperature
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
44/139
36
(0degF to 200degF ) and GPM (0 to 480) were randomly sampled to form a normal
distribution as shown in Figure 24.
Mean = 100, Standard Deviation = 80
Mean = 250, Standard Deviation = 30
Figure 24: Normal distribution for input Temperature and GPM
Figure 25: PDF and CDF of NPSHA1output
Assumptions made were based on expert opinions that suggested a NPSHA1
output range of 6.5 to 35. Monte Carlo simulation was used to generate this output
assuming a normal distribution of mean 23 and standard deviation of 3.6. ( Refer to
Figure 25).
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
45/139
37
Note that if the mathematical model were available, the two input distributions
can be fitted into the equation to generate a NPSHA1 output which follows a normal
distribution. The equation of NPSHA1 is the same as the NPSHA equation [3] in
section 5.1, except that the parameter is removed since it is considered to be zero.
In reality, it is more often than not that the mathematical Model of a system is
usually not available, and this is especially true for new systems which still lack
established model testing. BBN is therefore a useful tool to estimate the output of the
system.
Figure 26: Histogram of NPSHA1 to estimate the probability of A to L
The normal distribution of NPSHA1 is divided equally into 12 columns as
shown in the histogram of Figure 26. The histogram is aligned to the state A to L of
the DMLD as shown in Figure 7 assuming no overlap and uncertainty between the
states. The probabilities of A to L estimated based on the CDF of NPSHA1 ( Figure
25) are tabulated in Table 5:
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
46/139
38
Table 5: Probability data and value of A to L based on the CDF of NPSHA1
Gmax 23.06 2.28E-01
Hmax 25.414 2.53E-01
Imax 27.767 1.86E-01Jmax 30.12 9.06E-02
Kmax 32.473 2.91E-02
Lmax 34.826 6.19E-03
NPSHA1 min 6.59
The probabilities of GPM and Temperature inputs as shown in Table 6 are
estimated based on the CDF of inputs GPM and Temperature.
Table 6: Probability data of GPM and Temperature inputs
Input Symbol Range Probability
GPM HIGH GHI391-480
2.79E-02
GPM MID HIGH GMH301-390
1.95E-01
GPM MID GMD 181-300 5.51E-01
GPM MID LOW GML91-180
1.96E-01
GPM LOW GLW1 – 90
2.81E-02
GPM ZERO GZE 0 1.29E-03
TEMP LOW TLW 0 – 100 4.82E-01
TEMP HIGH THI 100 - 200 5.18E-01
In the real world, uncertainties are inevitable, and it is not realistic to represent
the output of GPM and Temperature based on the NPSHA’s DMLD structure.
Instead, it is more feasible to spread the outputs of GPM and Temperature over a
range of values represented by a distribution.
NPSHA1 Probability
Amax 8.9431 2.12E-05
Bmax 11.296 2.70E-04Cmax 13.649 2.38E-03
Dmax 16.002 1.39E-02
Emax 18.355 5.34E-02
Fmax 20.708 1.36E-01
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
47/139
39
To prove that overlaps do exist over the A to L states of NPSHA1 output, the
two extreme ends of each GPM and Temperature category are substituted into the
physical equation of NPSHA1. The minimum/maximum GPM and Temperature
values are then tabulated in Table 7:
Table 7: Representation of GPM and Temperature variables
Symbol Temp TEMP THI2 HI_max 200
THI1 HI_min 101
TLW2 LW_max 100
TLW1 LW_min 0
The minimum/maximum values of A to L tabulated in Table 8 are computed
by subsituting minimum/maximum GPM and Temperature into the mathematical
model.
For example, based on the DMLD structure:
A1 35.18 6.4 10 GHI1 0.085 THI2
A1 35.18 6.4 10 480 0.085 200 3.4344
Symbol GPM Value GHI1 HI_max 480
GHI2 HI_min 391
GMH1 MH_max 390
GMH2 MH_min 301
GMD1 MD_max 300
GMD2 MD_min 181
GML1 ML_max 180
GML2 ML_min 91
GLW1 LW_max 90
GLW2 LW_min 1
GZE ZE 0
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
48/139
40
A2 35.18 6.4 10 GHI2 0.085 THI1
A2 35.18 6.4 10 391 0.085 101 16.8106
Symbol NPSHA1 Value A1 A_min 3.4344
A2 A_max 16.81062
B1 B_min 8.4456
B2 B_max 20.79654
C1 C_min 12.42
C2 C_max 24.4983
D1 D_min 16.1064
D2 D_max 26.06502
E1 E_min 17.6616
E2 E_max 26.59494
F1 F_min 18.18
F2 F_max 26.595
Symbol NPSHA1 Value G1 G_min 11.9344
G2 G_max 25.39562
H1 H_min 16.9456
H2 H_max 29.38154
I1 I_min 20.92
I2 I_max 33.0833
J1 J_min 24.6064
J2 J_max 34.65002
K1 K_min 26.1616
K2 K_max 35.17994
L1 L_min 26.68
L2 L_max 35.18
From Table 8, note that the overlaps between A and L imply that uncertainties
do exist between the A to L states.
The DMLD of the pump system can be illustrated by a BBN as shown in
Figure 27. For this particular example, the area of focus would be on the comparison
of NPSHA1 (highlighted in red ). The software used to develop the BBN is IRIS [13].
Table 8: Numerical values of A to L
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
49/139
41
Figure 27: BBN interpretation of the pump system
In the perfect world, NPSHA1 can be represented based on the following
Bayesian model:
Pr 1 | Pr
Pr 1 | 1 2
Pr …
Pr 1 | 1 2
Pr
Pr 1 | Pr
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
50/139
42
This computation assumes no uncertainty between A and L. The probability
data of GPM and Temperature as shown in Table 6 would be entered into the GPM
and Temp node respectively. The conditional probability of A to L entered into the
NPSHA1 node is shown in Figure 28. Interpretations based on these conditions are
incorrect as there are overlaps in the NPSHA1 output data.
Figure 28: Conditional Probability table assuming no uncertainty between A to L
On the other hand, consider the case where the outputs of GPM and
Temperature are spread over a range of NPSHA1 distribution. The 12 output
combinations of GPM and Temperature are listed as follows:
Set1 represents the NPSHA1 output when GPM is HI and Temp is HI
Set2 represents the NPSHA1 output when GPM is MH and Temp is HI
Set3 represents the NPSHA1 output when GPM is MD and Temp is HI
Set4 represents the NPSHA1 output when GPM is ML and Temp is HI
Set5 represents the NPSHA1 output when GPM is LW and Temp is HI
Set6 represents the NPSHA1 output when GPM is ZE and Temp is HI
Set7 represents the NPSHA1 output when GPM is HI and Temp is LW
Set8 represents the NPSHA1 output when GPM is MH and Temp is LW
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
51/139
43
Set9 represents the NPSHA1 output when GPM is MD and Temp is LW
Set10 represents the NPSHA1 output when GPM is ML and Temp is LW
Set11 represents the NPSHA1 output when GPM is LW and Temp is LW
Set12 represents the NPSHA1 output when GPM is ZE and Temp is LW
Assume that all 12 sets follow a normal distribution with the corresponding
estimated means and standard deviations that are listed in Table 9. The distributions
between these sets have overlaps where more specifically, set 1 follows a left sided
truncated normal distribution, and set 12 follows a right sided truncated normal
distribution. The overlapped areas for each set would be summed up according to the
NPSHA1 groups.
Table 9: Mean and Standard Deviation of 12 sets
Sets Mean SD 1 10 2
2
14
2
3 18 2
4 21 1.5
5 22 1.2
6 23 1.2
Sets Mean SD 7 19 2
8 23 2
9 27 2
10 29 2
11 30 2
12 31 1.5
MATLAB was used to compute the weights for the states A to L for each set.
To illustrate this method, the derivation of Sets 1 and 2 would be explained in greater
depth as follows.
Truncated normal distribution
The PDF of the truncated normal distribution is represented by the equation:
| ,, ,
1
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
52/139
44
Where X ~ N (μ,σ2), X ϵ (a,b), -∞ ≤ a < b ≤ ∞. If b → ∞, then = 1; and if a → -∞, then = 0.
The CDF of the truncated normal distribution is represented by the equation:
| ,, ,
Similarly, if b → ∞, then = 1; and if a → -∞, then = 0.
The range of set 1 is first estimated, which is approximately between 3 and 17;
and represented by a normal distribution. Since set 1 is left truncated at a = 6.59 and b
→ ∞, the PDF of set 1 is given by:
| , , 6.59,∞ 1
1 6.59
Similarly, the CDF of this set is given by:
| , , 6.59,∞
6.59
1 6.59
where the respective μ and σ values can be obtained from Table 9.
The PDF plot of set 1 is shown in Figure 29. The blue plot represents the
normal distribution while the red line represents the truncated normal distribution.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
53/139
45
Figure 29: PDF of Set 1 which represents the GPM_HI and Temp_HI
Figure 30: CDF of Set 1 which represents the GPM_HI and Temp_HI
6.59
6.59
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
54/139
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
55/139
47
Figure 32: CDF of set 2 which represents the GPM_MH and Temp_HI
The range of set 2 takes on parameters A to G (referring to Table 4), and is
represented by B1 to B7 respectively. The probabilities of B1 to B7 are as follows:
Pr (B1) = 5.55E-03
Pr (B2) = 8.66E-02
Pr (B3) = 3.42E-01
Pr (B4) = 4.06E-01
Pr (B5) = 1.44E-01
Pr (B6) = 1.49E-02
Pr (B7) = 4.46E-4
The same method is then used to estimate sets 3 to 11. Set 12 follows a similar
methodology as set 1, except that this time it is represented by a truncated normal
distribution that is right truncated at b = 34.826 and a → -∞.
The PDF of set 12 is given by the equation:
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
56/139
48
| , ,∞, 34.38 1
34.38
Similarly, the CDF of set 12 is given by:
| ,,∞, 34.38 34.38
where the respective μ and σ values can be obtained from Table 8.
Figure 33: PDF of set 1 which represents the GPM_ZE and Temp_LW
Figure 33 illustrates the graphical representation of PDF of set 12. The blue
plot represents the normal distribution while the red line represents the truncated
normal distribution. Note that there is only a slight truncation to the right that results
in a small change in the PDF.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
57/139
49
Figure 34: CDF of set 12 which represents the GPM_ZE and Temp_LW
The CDF of set 12 is represented by the graph in Figure 34. The blue plot
represents the normal distribution while the red line represents the truncated normal
distribution. After computing the probability distributions for all 12 sets, the
probability of the weights of each set can be estimated. The results are entered into the
CPT as shown in Figure 35.
Figure 35: Conditional probabilities of A to L
0 0 0 0 0 0 0 0
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
58/139
50
S1 = Pr (GHI) ∩ Pr (THI)
S2 = Pr (GMH) ∩ Pr (THI)
S3 = Pr (GMD) ∩ Pr (THI)
S4 = Pr (GML) ∩ Pr (THI)
S5 = Pr (GLW) ∩ Pr (THI)
S6 = Pr (GZE) ∩ Pr (TLW)
S7 = Pr (GHI) ∩ Pr (TLW)
S8 = Pr (GMH) ∩ Pr (TLW)
S9 = Pr (GMD) ∩ Pr (TLW)
S10 = Pr (GML) ∩ Pr (TLW)
S11 = Pr (GLW) ∩ Pr (TLW)
S12 = Pr (GZE) ∩ Pr (TLW)
Note that the probability values of GPM and Temp can be found in Table 6.
The probabilities of A to L can be estimated by adding up the overlaps. For example,
Pr (A) = A1 * S1 + B1 * S2
= 2.63E-01 * 1.44E-02 + 5.55E-03 * 1.01E-01 = 4.37 E -3
Pr (B) = A2 * S1 + B2 * S2 + C2 * S3 + G2 * S7
= 4.66E-01 * 1.44E-02 + 8.66E-02 * 1.01E-01 + 3.44E-04 * 2.85E-01 + 4.43E-5 *
1.34E-02 = 1.56E-02
Applying the same calculations to the rest of the parameters, Table 10 lists the
estimated probabilities of A to L using the BBN structure shown in Figure 27.
Symbol Value
S1 1.44E-02
S2 1.01E-01
S3 2.85E-01S4 1.01E-01
S5 1.45E-02
S6 6.71E-04
S7 1.34E-02
S8 9.39E-02
S9 2.65E-01
S10 9.43E-02
S11 1.35E-02
S12 6.24E-04
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
59/139
51
Table 10: Probability of A to L
Symbols Probability Cumulative A 4.37E‐03 4.37E‐03
B 1.56E‐02 2.00E‐02
C 4.18E‐02 6.18E‐02
D 8.25E‐02 1.44E‐01
E 1.42E‐01 2.86E‐01
F 1.58E‐01 4.44E‐01
G
1.28E‐
01
5.72E‐
01
H 1.03E‐01 6.75E‐01
I 1.50E‐01 8.25E‐01
J 1.26E‐01 9.51E‐01
K 4.32E‐02 9.94E‐01
L 6.16E‐03 1.00E+00
The following chapter will compare and analyze the results obtained via the
two methodologies. The respective pros and cons of both BBN and FLM will also be
discussed.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
60/139
52
Chapter 6: Findings and Discussion
This section discusses the probability data of NPSHA1 obtained from the
BBN model based on the assumption of whether uncertainty is incorporated into
constructing the BBN. The probability data and NPSHA1 data are tabulated in Table
10.
Comparisons will also be made on both BBN model and FLM with respect to
the reference model which for this case is the mathematical model. The criteria for
comparisons are as follows:
• Accuracy of results*
• Resolution of data*
• Flexibility in model adjustment
• Ease of building the model
• Ability to update the model
• Requirement for precise data
• Mathematical strength
• Areas of application
The strengths and weaknesses of the two methodologies are also discussed as
a follow up to the comparisons.
* Note that some of the comparisons are made in the context of the pump system
application.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
61/139
53
6.1 Discussion on Bayesian Belief Network Probability Data
In the comparison of the probability data of NPSHA1, the assumption of
whether uncertainty is incorporated gives rise to the following two scenarios (Refer to
Table 11 for the comparisons of probability data of NPSHA1 based on the two
scenarios):
Scenario 1: Bayesian Belief Network with no uncertainty
Bayesian Belief Network methodology as discussed in section 5.3 was used to
estimate the NPSHA1 output, with no physical equation available. The input
parameters mapped to the NPSHA1 output range from A to L via a DMLD as shown
in Figure 7. IRIS [13] was used to model the BBN representation of the pump system.
The CPT as shown in Figure 28 was set up with the assumption of no uncertainty
between the outputs, given the GPM and Temperature input conditions.
Analyzing the “Probability with No Uncertainty” column in Table 11, the
probabilities of F and L are very low, suggesting that it is unlikely that the range of
NPSHA will fall in state F and state L. Further observation of the expected NPSHA1
output (refer to “Midpoint of NPSHA1a Interval” column) between D and G reveals
that the data are very close to each other which might raise some concern over the
credibility of the results obtained.
Since no uncertainty is assumed, the midpoint of NPSHA1 interval outputs for
each state from A to L are point estimates. This interpretation is not realistic as data
sources often lack precision, and consequently rarely produces point estimate results.
Given that there are uncertainties within the input parameters, one should also expect
uncertainties in the outputs of A to L. This issue is addressed in scenario 2.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
62/139
54
Scenario 2: Bayesian Belief Network considering uncertainty
Using BBN methodology can incorporate uncertainties for each of the output
A to L. This method assumes a normal distribution with estimated means and standard
deviations for A to L (refer to Table 9). The CPT, obtained in Figure 32 shows a
distribution for each GPM/Temp alternative. The summation of the overlaps between
the distributions of the input alternatives provides a good estimate of the probabilities
of A to L.
Analyzing the “Probability with Uncertainty” column in Table 11, the
probabilities of A to L are considerably evenly distributed, with the two ends of the
tail, A and L having smaller probabilities as compared to the rest. The range of
NPSHA1 between A to L are well spread out, indicating uncertainties between each
range. The results obtained from the assumption of uncertainty are more realistic and
would be used for further discussion in section 6.2.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
63/139
55
Table 11: Comparison of probability data of NPSHA1 based on 2 scenarios
Temp GPM
Probability
with No
Uncertainty Cumulative
Midpoint+ of
NPSHA1a
Interval
Probability
with
Uncertainty Cu
A 100-200 391-480 1.64E-02 1.64E-02 8.96 4.37E-03
B 100-200 301-390 1.03E-01 1.19E-01 16.64 1.56E-02
C 100-200 181-300 2.76E-01 3.95E-01 20.00 4.18E-02
D 100-200 91-180 1.00E-01 4.95E-01 22.10 8.25E-02
E 100-200 1 to 90 1.66E-02 5.11E-01 22.64 1.42E-01
F 100-200 0 8.43E-04 5.12E-01 22.71 1.58E-01
G 0-100 391-480 1.56E-02 5.28E-01 22.79 1.28E-01
H 0-100 301-390 9.77E-02 6.26E-01 23.31 1.03E-01
I 0-100 181-300 2.62E-01 8.88E-01 25.37 1.50E-01
J 0-100 91-180 9.54E-02 9.83E-0128.64
1.26E-01
K 0-100 1 to 90 1.58E-02 9.99E-01 32.16 4.32E-02
L 0-100 0 8.02E-04 1.00E+00 34.72 6.16E-03
* NPSHA1b is normally distributed over the expected data. + Midpoint estimates used as the closest estimation for the mean of
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
64/139
56
6.2 Comparison of NPSHA1 Data with Reference Data
NPSHA1 output obtained via the BBN and FLM were compared to the
NPSHA1 output of the reference model over four different temperatures of 40, 80,
120 and 160 ͦ F. The results are tabulated in Tables 12 and 13.
6.2.1 Bayesian Belief Network Model Comparison and Discussion
Accuracy of results
The NPSHA1 estimated via BBN gives expected values over a range of
uncertainties. For example, given the assumption that GPM is zero and temperature is
low, NPSHA output is distributed across H to L. There is a 61% probability of being
in state K, which is estimated to be between the range of 26 and 36; and a 27%
probability of being in state J, which is estimated to be between 24 and 35 ( As shown
in Figure 36 ). Note that the probability of state K dominates the other states.
Figure 36: Distribution of NPSHA given GPM and Temperature evidence
Pr (0
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
65/139
57
Referring to Table 12 (highlighted in red ), when GPM=0 and Temperature
=40 ͦ F, the midpoint of NPSHA1 output interval via BBN is 32.18, and has an
uncertainty range of 26 to 36. Compare this result with the reference NPSHA of
31.78. One would notice that there is no uncertainty in the GPM input when GPM
equals to zero while on the other hand, uncertainty exists in the input Temperature
parameter. Comparing the two results, there is a difference of approximately 1.26%,
which provides a reasonable estimate.
Consider another scenario where uncertainties exist for both inputs GPM and
Temperature. Using the same methodology, this time given that GPM is mid low and
temperature is high, NPSHA1 output is distributed across D to I. There is a 38%
probability of being in state F, which is estimated to be between the range of 18 and
27; and a 49% probability of being in state G, which is estimated to be between 12
and 27. Note that the probabilities of NPSHA1 falling in state F and G are relatively
higher as compared to the previous scenario.
Referring to Table 12 (highlighted in red ), when GPM = 120 and T = 160 ͦ F,
the midpoint of NPSHA1 output interval via BBN is approximately 17.85, and has an
uncertainty range of 12 to 27. Compare this result with the reference NPSHA of
20.6584. Notice that the uncertainty range is larger, and the difference between the
expected NPSHA1 and the reference NPSHA1 is approximately 13.59%.
Total absolute error of the NPSHA1 data at Temperature = 40deg ͦF is
calculated to be 38.17, which was derived by aggregating all absolute error terms
within the temperature range as listed in Table 12. The total percentage error is
therefore given by 100 .. 100 8.87%. Using the same
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
66/139
58
computations for all error terms at Temperature = 160deg ͦF, the total percentage error
is found to be:.. 100 10.89%.
The relatively low percentage errors at the two temperature points suggest
that the results obtained via BBN on the two scenarios do provide a satisfactory
estimate of NPSHA1, although the extent of the errors are still dependent on the
degree of uncertainty of the input variables. In this particular application, the data
obtained from the reference model all falls within the uncertainty range specified by
the BBN model for every scenario. The important question is: what is the range that
would be considered acceptable? In order to have more precise results, the input
parameters need to be better defined such that uncertainties can be reduced.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
67/139
59
Table 12: Comparison of NPSHA1data between BBN model and Reference model at T = 40, 80, 120 and 160 ͦ F
GPM
T = 40 T = 80 T = 120 T = 160
REF BBN Error REF BBN Error REF BBN Error REF BBN Error
0 31.78 32.18 0.40 28.38 32.18 3.80 24.98 21.24 3.74 21.58 21.24 0.34
20 31.75 29.94 1.81 28.35 29.94 1.59 24.95 19.59 5.36 21.55 19.59 1.96
40 31.68 29.94 1.74 28.28 29.94 1.66 24.88 19.59 5.29 21.48 19.59 1.89
80 31.37 29.94 1.43 27.97 29.94 1.97 24.57 19.59 4.98 21.17 19.59 1.58
120 30.86 27.12 3.74 27.46 27.12 0.34 24.06 17.85 6.21 20.66 17.85 2.81
160 30.14 27.12 3.02 26.74 27.12 0.38 23.34 17.85 5.49 19.94 17.85 2.09
200 29.22 24.98 4.24 25.82 24.98 0.84 22.42 16.08 6.34 19.02 16.08 2.94
220 28.68 24.98 3.70 25.28 24.98 0.30 21.88 16.08 5.80 18.48 16.08 2.40
240 28.09 24.98 3.11 24.69 24.98 0.29 21.29 16.08 5.21 17.89 16.08 1.81
280 26.76 24.98 1.78 23.36 24.98 1.62 19.96 16.08 3.88 16.56 16.08 0.48
320 25.23 23.63 1.60 21.83 23.63 1.80 18.43 14.16 4.27 15.03 14.16 0.87
340 24.38 23.63 0.75 20.98 23.63 2.65 17.58 14.16 3.42 14.18 14.16 0.02
380 22.54 23.63 1.09 19.14 23.63 4.49 15.74 8.07 7.67 12.34 8.07 4.27400 21.54 22.57 1.03 18.14 22.57 4.43 14.74 8.07 6.67 11.34 8.07 3.27
440 19.39 22.57 3.18 15.99 22.57 6.58 12.59 8.07 4.52 9.19 8.07 1.12
480 17.03 22.57 5.54 13.63 22.57 8.94 10.23 8.07 2.16 6.83 8.07 1.24
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
68/139
60
Resolution of data
In this pump system example, the Temperature input parameter only assumes
dichotomous categories, namely: low (0-100 ͦ F) and high (101-200 ͦ F). Referring to
Table 12, the expected NPSHA1 outputs estimated via BBN are the same for
Temperature input 0 to 100 ͦ F; and 101 to 200 ͦ F respectively.
Similarly, GPM input is divided into 6 categories such that NPSHA1 output is
the same for every GPM input that falls within each category. Referring to Table 12,
GPM = 20, 40 and 80 all falls within GML category where this results in the same
expected NPSHA1 output of 31.469 with an uncertainty range between 20 to 32.
Therefore, such categorical classifications of parameters would reduce the sensitivity
of output results, and may fail to distinguish output values that correspond to the
different values of the same input category. Plotting GPM with respect to NPSHA1
would result in a discrete graph that represents a loss of resolution, instead of a
smooth curve as shown in Figure 8.
In order to improve the resolution, more conditions would need to be defined
in the input parameter nodes. This can be achieved by obtaining more information on
the input parameters As the CPT increases in size and complexity, more time would
be required to build the BBN model. In short, there is a corresponding increase in the
resolution of the output data with larger number of conditions defined for each input
node.
Flexibility in model adjustment
Representing the pump system as a BBN is a more flexible approach to solve
system problems. Consider a case where the NPSHA output is known and estimations
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
69/139
61
are necessary for the input parameters. In this instance, expert opinion estimates that
the NPSHA output is approximately between 13 and 17, which falls within state D.
Setting the evidence of NPSHA node of the BBN to D, the input parameters can be
estimated based on the graphical representation of BBN as shown in Figure 37.
Figure 37: BBN of pump system given evidence that NPSHA1 is D
In order to achieve NPSHA1 output that falls within state D, there is a 51%
probability that GPM is in the mid high range and 47% probability of being in the mid
range. There is a 98.8% probability that temperature would fall in the high region.
Ease of building the model
Although building the structure of a BBN model is not difficult, the process of
understanding, identifying and estimating the probability data for each condition of
Pr (100< T≤200 | 13
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
70/139
62
every node makes the modeling challenging and tedious. The model becomes
complicated when many uncertainties are being considered in the input parameters,
and overlapping data are expected within the output parameters.
Currently, there is no established method to incorporate BBN into a system
with cyclic network. This proves to be a major problem as many complex real life
systems consist of cyclic networks. To counter this limitation, Tang, Liu and Qian
[14] proposed using DBN model to solve complex real life system with feedback
loops.
To elaborate further, DBN is based on temporal time series data. Consider a
simple example [14] with a feedback loop A→B→C→A, where node C at time t has
an influence on node A at time t+1 ( Refer to Figure 38).
Figure 38: Dynamic Bayesian Network with feedback loop [11]
Dynamic Bayesian Network can construct a cyclic regulation by dividing the
states of a variable by time slices [15]. Node A at time t has an influence on itself at
t+1, i.e. as shown in the dotted lines in Figure 39. Node C in this case is primary time dependent where .
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
71/139
63
One recommendation is to implement DBN on a primary feedback loop if it
satisfies the primary time dependency requirements, i.e. Node C and ignore all trivial
time dependencies, i.e. Node A.
Figure 39: Time expansion of the dynamic network in Figure 34 [11]
In another paper, Z. Mohaghegh, R. Kazemi, and A. Mosleh [16] proposed
using hybrid modeling to address dynamic complex system with feedback loops. They
use simulation based techniques such as ABM [17] and SD [18] to model complex
model with impossible analytical solutions.
System Dynamics technique represents dynamic deterministic relations in
hybrid modeling environment and provides dynamic integration among various
modules/subsystems. These subsystems which can be modeled via various techniques
such as BBN would have their own inputs and outputs to the SD module.
Consequently, the entire hybrid model would have the relevant capability to capture
feedback loops and delays. The SD software used in this paper [16] was STELLA.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
72/139
64
For example, the hybrid model allows the targeted subsystem that is modeled
via BBN to be imported and processed inside SD which has delays and feedback
loops. The estimated value calculated from SD would then be exported back into the
BBN environment. Such integrated processes would pose as a good alternative to
resolve the challenges that arise from integrating BBN into systems with feedback
loops. Figure 40 shows a demonstration on how BBN can be integrated into a system
with feedback loops.
Figure 40: Training module within a SD environment [16]
This thesis implements the use of BBN on a simple two inputs - one output
system, to demonstrate that it is possible to model a dynamic system probabilistically.
However, incorporating uncertainty into a complex system using BBN is highly
challenging as uncertainty has to be considered in every parameter and condition.
Building a BBN on multiple hierarchies of complex systems would require much
more computational effort and time. Therefore such applications are often subjected
to constraints posed by the limited resources in the industrial context.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
73/139
65
Ability to update model
Bayesian Belief Network is a powerful tool for reasoning and learning with
uncertainty. Therefore the capability of Bayesian updating is a major advantage of
learning via BBN. For example, in the face of uncertainty in the NPSHA1 output
range, one can only approximate the mean and standard deviation over a certain form
of distribution. Assuming that the expected values of NPSHA1 follow a normal
distribution, and the likelihood model also follows a normal distribution with known
standard deviation, one can use conjugate priors to estimate the posterior mean of
NPSHA1, given that the initial estimates for mean and standard deviation are the prior
parameters.
Posterior mean is given by
∑
[19]
Where the evidence is failures at times , is the prior mean, is the priorstandard deviation, and = v
Posterior variance is given by 2 1102
[19]
Requirement for precise data
The findings reinforce the notion that the BBN model does not require exact
historical data or evidence to produce convincing results. As demonstrated in the
pump system application in section 5.3, precise evidences of the two input parameters
are not necessary to produce an accurate NPSHA1 output, even when the physical
equation is not available.
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
74/139
66
However accuracy of the output will be limited to the certainty of the input
parameters. Therefore, highly uncertain inputs are often accompanied by more
ambiguous outputs. Further research can look into the determination of the level of
uncertainties within the data that is deemed as acceptable.
Mathematical strength
Bayesian Belief Network is modeled after Bayes’ theorem which is a proven
and established mathematical model. As a result, BBN methodology is also well
documented and widely accepted among scholars and practitioners.
Areas of application
One rule of thumb in using BBN is that it is highly effective for modeling
applications where some information is already known and incoming data is uncertain
or partially unavailable. It is especially useful when historical or current information
is vague, incomplete, conflicting and uncertain [11].
However, there is a flip side to every coin as BBN is not adapted to work in
applications with time delays and feedback loops. Tang, Liu and Qian [14], and Z.
Mohaghegh, R. Kazemi, and A. Mosleh [16] have proposed DBN and Hybrid
modeling respectively to counter these problems. BBN is also useful in applications
where the exact mathematical model is difficult or impossible to be determined.
6.2.2 Fuzzy Logic Comparison and Discussion
Accuracy of results
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
75/139
67
Using the FLM methodology as discussed in section 5.2, the NPSHA1 fuzzy
data can be estimated via the plot as shown in Figure 18. From the results obtained
from Table 13, Figure 41 illustrates the comparison of NPSHA1 between reference
and FLM data at Temperature = 40 ͦ F.
Figure 41: Comparison of NPSHA at T=40 between Reference model and FLM
With reference to Figure 41, at Temperature = 40 ͦ F, the trend of fuzzy logic
line is consistent with the physical model line. It is not possible to obtain precise
results produced by the reference model due to the fuzziness or uncertainty defined
for the input and output parameters. The fuzzy output is more accurate towards the
two extreme ends of plot and less accurate in the middle. Referring to Table 13
(highlighted in red ), at Temperature =40 ͦ F and GPM = 120, the NPSHA1 Fuzzy
output is 27.67 as compared to the reference NPSHA1 of 30.86, where the difference
is approximately 10.3%. Referring to Table 13(highlighted in red ), at Temperature
Ref
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
76/139
68
=160 ͦ F and GPM=120, the NPSHA1 Fuzzy output is 14.33 as compared to the
reference NPSHA1 of 20.66, which translates into approximately 30% difference.
The total absolute error for NPSHA1 data at Temperature = 40deg ͦ F is 35.63,
which is obtained by summing up all absolute error terms at this temperature found in
Table 13.. The total percentage error is therefore calculated as
100 .. 100 8.28%. Similarly for Temperature = 160deg ͦ F, the total
percentage error is .
Assuming a fixed input GPM, as the temperature increases, the variation
between the fuzzy NPSHA1 and reference NPSHA1 becomes larger. Such inaccuracy
is due to the fact that input temperature has only two membership functions, resulting
in fuzzier outputs.
In a nutshell, fuzzy logic is tolerant of imprecise data. Fuzzy reasoning builds
this understanding into the process rather than tacking it at the end [2].
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
77/139
69
Table 13: Comparison of NPSHA1data between Fuzzy Logic model and Reference Model at T = 40, 80, 120 and 160 ͦ F
GPM
T = 40 T = 80 T = 120 T = 160
REF FUZZY Error REF FUZZY Error REF FUZZY Error REF FUZZY
0 31.78 33.69 1.91 28.38 29.60 1.22 24.98 24.40 0.58 21.58 20.31
20 31.75 31.24 0.51 28.35 26.35 2.01 24.95 24.64 0.31 21.55 19.77
40 31.68 30.59 1.09 28.28 26.78 1.50 24.88 22.23 2.65 21.48 18.57
80 31.37 30.81 0.56 27.97 26.57 1.40 24.57 21.43 3.14 21.17 17.19
120 30.86 27.67 3.19 27.46 23.60 3.86 24.06 18.40 5.66 20.66 14.33
160 30.14 27.67 2.47 26.74 23.60 3.14 23.34 18.40 4.94 19.94 14.33
200 29.22 24.45 4.77 25.82 20.15 5.67 22.42 15.85 6.57 19.02 11.55
220 28.68 24.90 3.78 25.28 20.57 4.71 21.88 15.43 6.45 18.48 11.10
240 28.09 24.98 3.12 24.69 20.57 4.13 21.29 15.43 5.86 17.89 11.02
280 26.76 24.45 2.32 23.36 20.15 3.21 19.96 15.85 4.11 16.56 11.55
320 25.23 21.12 4.11 21.83 16.83 5.00 18.43 13.17 5.26 15.03 8.88
340 24.38 21.67 2.71 20.98 17.60 3.38 17.58 12.40 5.18 14.18 8.33
380 22.54 20.71 1.83 19.14 16.70 2.44 15.74 12.17 3.56 12.34 8.20
400 21.54 20.41 1.13 18.14 15.89 2.25 14.74 12.00 2.74 11.34 7.77
440 19.39 19.22 0.17 15.99 14.98 1.01 12.59 11.18 1.41 9.19 6.76
480 17.03 18.98 1.94 13.63 14.57 0.93 10.23 9.43 0.80 6.83 5.02
8/13/2019 Bayesian Belief Network and Fuzzy Logic Modeling for Compariosion of Performance of Dynamic System
78/139
70
Resolution of data
Fuzzy data has better resolution as compared to the BBN model. Referring to
Figure 41, note that there is a decreasing trend in the fuzzy plot which is noticeably
consistent with the reference plot. In order to improve the resolution, the membership
functions for the input and output parameters should be broken down further into finer
details. As such, more if/then rules between the inputs and outputs would need to be
defined.
Flexibility in model adjustment
Fuzzy Logic Modeling offers higher flexibility in its ability to adjust and
change the boundaries and membership functions of both input and output parameters
with little effort. However, application is limited in situations where the designers
only have information regarding the output parameters, and will need to estimate an
unknown input parameter.
Ease of building the model
The process of adapting FLM into a complex system is relatively easy and
straightforward. FIS greatly depends on a number of conditional “if-then” rules in
order to model the system output. Although these rules are easy to write, if the
designers do not understand the system well, they would not be able to provide
sufficient rules to accurately describe the system.
Fuzzy logic can be adapted onto multiple hierarchies of complex systems by
using SIMULINK [20], which is a dynamic system simulator for MATLAB. It
supports linear and non linear systems, and can be modeled in continuous time,
sampled discrete time, or a hybrid of both. Using the graphical user interface
8/13/2019 Bayesian Belief Network and Fuzzy Logic